What we definitely know about endlessly possible

Published 4:00 am, Sunday, August 7, 2005

While science and religion often make for odd, if not hostile, bedfellows, at times they have been quite cozy indeed. Take the strange case of
Georg Cantor
. After being relegated to an obscure mid-tier
university
, blocked from leading journals and openly mocked by his peers, including his former mentor, the late 19th century German mathematician found refuge for his groundbreaking work on infinities in, of all places, the
Roman Catholic Church
.

Yes, the very same church that in 1600 burned philosopher Giordano Bruno at the stake for espousing his belief in an infinite universe was in Cantor's day seeking to reconcile science and religion through, among other things, repercussion-free dialogue. Catholic theologians welcomed Cantor's ideas, which provided a workable way of understanding mathematical infinities, as evidence that humans could grasp the infinite and could also, therefore, have a greater understanding of God, himself infinite.

What a welcome relief this must have been to the chronically depressed Cantor! As John D. Barrow writes in "The Infinite Book: A Short Guide to the Boundless, Timeless and Endless," Cantor "started to tell his friends that he had not been the inventor of the ideas about infinity that he had published. He was merely a mouthpiece, inspired by God to communicate parts of the mind of God to everyone else."

Today, Cantor's work on infinity is taken for granted by mathematicians, and religious authorities seem more wont to ban sound science from schools than foster it. But the story demonstrates the unusual position infinity occupies in intellectual history, broad enough in scope to engage the minds of philosophers, astronomers, physicists, theologians and literary authors alike. And not just in the past, but the present as well.

Barrow eloquently explains: "Infinity is a player of great significance who appears on the stage only when the crucial questions of existence are raised. Infinity offers its services when we seek to know if the Universe began or whether it will ever end, whether life will always be part of its landscape, and whether there are tasks which can never be accomplished. Infinity challenges us to contemplate the duplication of ourselves and all that we hold dear, and to ponder the cogency of all possibilities, potential and actual. It undermines our sense of the precious by suggesting a randomly infinite universe will eventually conjure up the works of Shakespeare, somewhere, as if created by a regiment of monkeys armed with typewriters. Infinity also seeks to guard us from taking the wrong path in our quest to unravel the deepest of Nature's secrets about the ultimate structure of mass and energy."

Barrow, a mathematician and popular science writer who in previous books tackled "Theories of Everything" and the scientifically impossible, captures many of humanity's attempts to grasp the ungraspable in this engaging read. "The Infinite Book" starts with a quick tour of early philosophers' crude attempts to define, dismiss or otherwise contain the infinite. Without telescopes, microscopes, calculators or other modern tools to guide them, these thinkers' understanding of the world was informed by a sense of their own limits and a belief in the transcendent.

Their views on the infinite reflected this. Aristotle dealt with the paradoxes of infinity by dividing it into actual and potential infinities, the former being impossible. A sequence of numbers could go on forever but, as nobody could ever count it, could not exist in actuality. St. Augustine posited that what seemed infinite to humans was finite to God. In the 17th century, Pascal believed that humans encountered infinities regularly, but could never fully grasp them. Eighteenth century philosopher Immanuel Kant took a similar path, arguing that infinities existed, but humans' perceptive powers were too limited to perceive them.

Not being able to comprehend infinity didn't stop people from trying and eventually succeeding. Galileo described some of mathematical infinity's most troubling paradoxes, such as the fact that an infinite sequence of whole numbers (one, two, three, four, five ...) and an infinite sequence of even numbers (two, four, six, eight ...) would be the same size even though the latter would appear to contain far fewer numbers. It was Cantor who made sense of such conundrums, dividing infinities into the countable, which could be put into a one-to-one correspondence with a list of natural numbers, as Galileo's could, and uncountable, which could not. He showed that mathematical infinities could, in fact, exist and be understood. Some infinities were even bigger than others. Catholic theologian Constantin Gutberlet, Barrow writes, "responded by seizing upon Cantor's mathematical work to argue that it provided clear evidence that the human mind could contemplate the actual infinite," and therefore "get closer to the true nature of the Divine."

Of course, as Barrow is quick to point out, "infinity is not a big number. " It behaves quite differently and is riddled with problems that can lead the imagination to some very unsettling places. For instance, the infinite replication paradox, which the author sums up succinctly as such: "In a universe of infinite size, anything that has a non-zero probability of occurring must occur infinitely often. Thus at any instant of time, for example the present moment, there must be an infinite number of identical copies of each of us doing precisely what each of us is now doing." Yikes! In such examples one can see why, when it comes to infinities, science and the "softer" pursuits of theology, philosophy and ethics -- to say nothing of literature -- to this day frequently find themselves traversing the same territory.

It is also in the midst of such mixed company that we find the agile mind of Barrow at its best: moving from science's attempts to evade the implications of the infinite replication paradox to converse with the works of Jorge Luis Borges that embrace them; pausing to acknowledge St. Augustine's worry that multiple worlds would require multiple crucifixions; as well as paying his respects to Friedrich Nietzsche, who, Barrow writes, argued "that we should act as if we knew our actions would be infinitely repeated." One only wishes that Barrow would escape the clutches of the Western canon every now and then to dialogue with Eastern philosophy and religion's own fascinating takes on the infinite.

That's one quibble with this book. Here's another. While Barrow's remarkable ability to provide clear, concise, engaging and distinctly finite explanations -- even when describing some fairly advanced concepts -- will, for the most part, be much appreciated, this approach occasionally breaks down and mathematical equations like L'=L(1-V²/c²){+ 1/2} get tossed onto the page without much explanation, as though they read as clearly as 1 + 2 = 3. Readers without the proper background in science and math will probably find themselves feeling a bit left behind at times. At the same time, armchair cosmologists who've pored over every book and theory may find some of Barrow's more scientific material a bit too familiar. This is the fine line every popular-science writer must walk. Fortunately, Barrows maintains his balance more often than not.

So where does all this talk of infinity leave us? Right back where we started, of course, smack dab in the land of conjecture or, specifically, the question of eternal life. In Barrow's exploration of this final topic, we see clearly science's uncanny ability to provide many answers (but no Answer) juxtaposed against religion's ability to supply a ready Answer but no real answers. How appropriate, really, that from the mixed roots of the infinite should spring up so many unanswerables.

Rather than pick a side, Barrow seems content throughout this book to simply acknowledge these, in one case ending a chapter on whether there can be other planets and universes with the quizzically titled subsection, "How should we then live?" How indeed! And when he writes, "You can discover whether the Universe is infinite, but the learning will take an infinite time, " one gets the eerie feeling that we haven't progressed as far from Aristotle's conjectures as we'd like to think.

But this, too, is the charm of the infinite: that one can learn so much about it yet know so little, can finish a book, but never reach the subject's end.

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